Suppose $\Omega$ is a bounded domain in Euclidean space and you can take it as convex as you wish. Is there a good place to look for a bunch of barrier type functions.
In particular I am looking for a bunch of examples so I can try and show a boundary estimate on the gradient of a harmonic function.
I am attempting to come up with some (what I think) are typical ones; but I would prefer to just look in a text book if they have a bunch of standard ones.
Edit. I will give some more details and a potential barrier and comments on whether this is nonsense or not are greatly appreciated (my sense of direction and geometry is gone and sorry if this is a trivial question). Assume $\Omega$ is in $R^N$ with smooth boundary and $ \Omega \subset R^N_+$ with $ 0 \in \partial \Omega$ and we assume its strictly convex. We write $ x=(x', x_N)$ and we suppose $g$ is smooth on the boundary of $\Omega$. Suppose also that $\Omega$ sits inside a ball with center $x^1$ and radius $R$ which is contained in the upper half space with $ 0 \in \partial B(x^1,R)$. Can we show for large enough $C$ (just depending on smoothness of $g$ and $\Omega$ that $$C (R^2 - |x-x^1|^2) + g(0) + \nabla_{x'} g(0,0) \cdot x' + C x_N \ge g(x)$$ for all $ x \in \partial \Omega$. Of course away from the origin there is no issue; but my sense of direction is not strong enough to see if this holds near the origin. I removed (I think) the linear term of $g$ which might have been causing trouble but I am not sure this is sufficient or .
